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It is clear to me that if I take the Schwarzschild metric $$ds^2 = \left(1-\frac{2M}{r}\right)dt^2 - \left(1-\frac{2M}{r}\right)^{-1} dr^2$$ and choose $\rho = 2\sqrt{\frac{r}{2M} -1}$ then I get the Rindler coordinates when $\rho << 1$.$$ ds^2 = \frac{\rho^2}{4} dt^2 - 4M^2 d\rho^2$$ Therefore I can say that when I am close to the horizon of a black hole at constant $\rho$ (or $r$ equivalently) then I am constantly accelerating outwards and do my space-time diagrams in Rindler coordinates. But this approximation is not valid when $\rho >> 1$, i.e. when the field is weak.

In the weak field approximation we have $$ds^2 = \left(1-\frac{2M}{r}\right)dt^2 - \left(1+\frac{2M}{r}\right) dr^2$$ my intuition tells me that to stay at $r$ constant I have to be constantly accelerating outwards, but I can't find any coordinate transform that brings me to the Rindler coordinates from the weak field approximation. Please help :)

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  • $\begingroup$ It does not work in the exact metric, only if you take $r$ close to $2M$ (or equivalently $\rho << 1$) you end up with the Rindler metric at first order. $\endgroup$ – albe_rola Jan 27 '19 at 23:37
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Before looking for a solution, we should think carefully about what kind of result we should expect. A metric of the form $$ \rho^2 dt^2- d\rho^2 \tag{1} $$ represents flat spacetime, and any worldline with constant $\rho$ is undergoing constant acceleration (constant weight). Because it is flat, a metric of this form can't be a good approximation to the Schwarzschild metric for all $\rho$. We can only expect it to be a good approximation for $\rho=\rho_0+\epsilon$ for some given $\rho_0$ and with sufficiently small $\epsilon$. Near the horizon, we have such an approximation with $\rho_0=0$; the hovering observer's acceleration diverges at the horizon, as it does in flat spacetime at the Rindler horizon. But away from the horizon, we will have $\rho_0\neq 0$, and then we can only expect a flat-spacetime approximation (1) to be valid modulo terms of order $\epsilon^2$.

With that in mind, start with the Schwarzschild metric in coordinates $t$ and $\mathbf{x}\equiv (x,y,z)$: $$ d\tau^2=A(r)dt^2-\frac{dr^2}{A(r)}-(d\mathbf{x}^2-dr^2) \\ r\equiv\sqrt{x^2+y^2+z^2} \hskip2cm A(r)\equiv 1-\frac{2M}{r}. \tag{3} $$ Let $\mathbf{x}=(0,0,z_0)$ be the worldline of the hovering observer, and expand everything to first order in $x$, $y$, and $\delta z\equiv z-z_0$. This gives $$ r^2\approx z_0^2+2z_0\delta z \hskip1cm \Rightarrow \hskip1cm r\approx z_0+\delta z=z \tag{4} $$ and $$ A(r)\approx 1-\frac{2M}{z}\approx A(r_0)+\frac{2M}{z_0^2}\delta z \equiv a+bz. \tag{5} $$ All approximations are understood to be valid modulo terms of order $(\delta z)^2$. Use these in (3) to get $$ d\tau^2\approx (a+bz)dt^2-\frac{dz^2}{a+bz}-dx^2-dy^2. \tag{6} $$ Now define $\rho$ by $$ \rho^2\equiv a+bz \tag{7} $$ to get the Rindler-like form $$ d\tau^2\approx \rho^2 dt^2-\frac{d\rho^2}{(b/2)^2}-dx^2-dy^2, \tag{8} $$ with the understanding that this is only valid in a neighborhood of $\mathbf{x}= (0,0,z_0)$, which corresponds to $\rho_0=\sqrt{a+bz_0}$. The approximation is good modulo terms of order $\epsilon^2$, where $\epsilon\equiv\rho-\rho_0$. This is the desired result.

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